Lecture 2829 FM Frequency Modulation PM Phase Modulation

1Lecture 2829FM Frequency ModulationPM Phase ModulationEE445102FM and PMDp is the phase sensitivity or phase modulation constant3FM and PMfor FM:Relationship between mf(t) and mp(t):4Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–8 Angle modulator circuits. RFC = radiofrequency choke.5FM and PM6Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–9 FM with a sinusoidal baseband modulating signal.7FM and PM8FM and PM differencesθ (t) = Dpm(t) ⇒ phase is proportional to m(t)instantaneous frequencydeviation from thecarrier is proportional to m(t)θ (t) = D f ∫−t∞ m(α)dαPM:FM:fi(t) − fc = D f m(t) ⇒voltHzD KvoltradiansD Kf fp p= ⇒= ⇒ModulationConstants9FM from PMPM from FM10FM from PMPM from FM11FM and PM SignalsMaximum phase deviation in PM:Maximum frequency deviation in FM:12ExampleLetFor PMFor FMDefine the modulation indices:13ExampleDefine the modulation indices:14Sine Wave ExampleThen15Spectrum Characteristics of FM• FMPM is exponential modulationLetRe( ) (2 sin(2 ))( ) cos(2 sin(2 ))j f t f tcc c mA e c mu t A f t f tπ β ππ β π+== +φ(t) = β sin(2πfmt)u(t) is periodic in fmwe may therefore use the Fourier series16Spectrum Characteristics of FM• FMPM is exponential modulationRe( ) (2 sin(2 ))( ) cos(2 sin(2 ))j f t f tcc c mA e c mc t A f t f tπ β ππ β π+== +c(t) is periodic in fmwe may therefore use the Fourier series17Spectrum Characteristics withSinusoidal Modulationu(t) is periodic in fmwe may therefore use the Fourier series18Jn Bessel Function19Jn Bessel Function20Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.stop 32921Lecture 29FM Frequency ModulationPM Phase Modulation(continued)EE4451022Narrowband FM•Only the Jo and J1 terms are significant•Same Bandwidth as AM•Using Eulers identity, and φ(t) fm, increasing fm does not increase Bc much•Bc is linear with fm for PM26Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.27Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.28Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation forvarious modulation indexes.29When m(t) is a sum of sine waves30When m(t) is a sum of sine waves31Sideband PowerSignal Amplitude: Ac := 1VModulating frequency: fm := 1KHzCarrier peak deveation: Δf 2.4KHz :=Modulation index: β Δffm:=β = 2.4Reference equation: x t ( )− ∞∞nA∑ ⎡ ⎣ c⋅Jn n ( ) ,β ⋅cos⎡ ⎣( ) ωc + n⋅ωm ⋅t⎤ ⎦⎤ ⎦=Power in the signal: PcAc22 1 ⋅ Ω:= Pc = 0.5 WCarsons rule bandwidth: BW 2 := ⋅( ) β + 1 ⋅fm BW 6.8 10 × 3 1s=Order of significant sidbands predicted by Carsons rule: n round := ( ) β + 1n 3 =Power as a function of number of sidebands: Psum( ) k− kknA( ) c⋅Jn n ( ) ,β 2∑ 2 1 ⋅ Ω=:=Percent of power predicted by Carsons rule:Psum( ) nPc⋅100 = 99.11832Sideband Power0 0.5 1 1.5 2 2.5 3050100PERCENT OF TOTAL POWERPsum( ) kPc⋅100k33Sideband Powerk 0 10 := ..Jk := β Jn k ( ) ,β = 2.4Pk := ( ) Jk 2n 3 =P0 21nj∑ Pj=+ ⋅ = 0.991J00 1 2 3 4 5 6 7 8 9102.508·1030.520.4310.1980.0640.0163.367·1035.927·1049.076·1051.23·1051.496·106= P00 1 2 3 4 5 6 7 8 9106.288·1060.2710.1860.0394.135·1032.638·1041.134·1053.513·1078.237·1091.513·10102.238·1012=34Sideband Powerj 0 5 := β .. := 0.1n 1 :=Vj := Jn j ( ) ,βUj := ( ) Vj 2U0 21nj∑ Uj=+ ⋅ = 1V0.9980.051.249 10 × − 32.082 10 × − 52.603 10 × − 72.603 10 × − 9⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠= U0.9952.494 10 × − 31.56 10 × − 64.335 10 × − 106.775 10 × − 140⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=35Sideband Powerβ := 0.6 n 1 :=Wj := Jn j ( ) ,β Xj := ( ) Wj 2X0 21nj∑ Xj=+ ⋅ = 0.996W0.9120.2870.0444.4 10 × − 33.315 10 × − 41.995 10 × − 5⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠= X0.8320.0821.907 10 × − 31.936 10 × − 51.099 10 × − 73.979 10 × − 10⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=36Bandwidth 18 = Modulation_index 7.9 =f fB f ( ) := δ⎡ ⎣f f , c + ( ) Fm n 0 + ⋅ ⎤ ⎦ + δ( ) f f , c − n Fm ⋅ := c − ( ) Fm n 1 + ⋅ ,( ) fc − n Fm ⋅ ..⎡ ⎣fc + ( ) Fm n 1 + ⋅ ⎤ ⎦Si f ( ) Ac ( ) J0 M ( ) ⋅δ( ) f f , c1nk∑ ⎡ ⎣Jn k M ( ) , ⋅δ⎡ ⎣f f ,( ) c + k Fm ⋅ ⎤ ⎦ + ( ) −1 k⋅ δ Jn k M ( ) , ⋅ ⎡ ⎣f f ,( ) c − k Fm ⋅ ⎤ ⎦⎤ ⎦=+⎡⎢⎢⎣⎤⎥⎥⎦:= ⋅n 9 = Bandwidth 2 n := ⋅ ⋅Fm Modulation_index M :=n roundM 1 := ( ) + 2 n is the number of significant sidebands per Carsons ruleMx10:=Fm 10 := 0 Modulating frequency single sinewavefc:= 0 10 ⋅ 4Ac:= 179FMPM modulation index : set toπ2 for peakphase dev of π2set toΔffm for frequency modulation. spectruis the same for sinewavemodulation.filename: fmsidebands.mcdavo 092104last edit date:2270737M=.4, Sideband Level =M2 for Narrowband FM0.2 2 1 0 1 200.20.40.60.8SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index38M=.9, Sideband Level =M2 for Narrowband FM0.5 3 2 1 0 1 2 300.51SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index39M=2.4, Carrier Null0.6 4 2 0 2 40.40.200.20.40.6SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index40M=3.8, first sideband null0.6 6 4 2 0 2 4 60.40.200.20.40.6SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index41M=5.1, second sideband null0.4 8 6 4 2 0 2 4 6 80.200.20.4SpectrumSingle Sided SpectrumPeak Volts1 0 2 4 6 8 100.80.60.40.200.20.40.60.81Carrier J01st Sidebands J12nd Sidebands J2Bessel FunctionsModulation_index42Power vs BW, M=0.1second term includes power in +Jnand power in Jn, i.e the upper andlower sideband pairsP M n ( ) , J0 M ( )221nk∑ Jn k M ( ) , 2=+⎛⎜⎜⎝⎞⎟⎟⎠:=99.999651 1.2 1.4 1.6 1.899.999799.9997599.999899.9998599.999999.99995% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 0.1 =Fm 1 = HzBandwidth 2 = HzP M n ( ) ,Ac22⋅100 = 10043Power vs BW, M=0.9⎝ ⎠98 1 1.5 2 2.5 3 3.5 498.59999.5100% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 0.9 =Fm 1 = HzBandwidth 4 = HzP M n ( ) ,Ac22⋅100 = 99.95844Power vs BW, M=2.4⎝ ⎠50 1 2 3 4 5 6 7 860708090100% power vs bandwidthNumber of Sideband pairsP M k ( ) ,Ac22⋅100nkM 2.4 =Fm 1 = HzBandwidth 8 = HzP M n ( ) ,Ac22⋅100 = 99.94545Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–16 Anglemodulated system with preemphasis and deemphasis.46Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0131424920Figure 5–16 Anglemodulated system with preemphasis and deemphasis.47AM vs FM• FM capture effect: A phenomenon, associated with FMreception, in which only the stronger of two signals at ornear the same frequency will be demodulated.– The complete suppression of the weaker signal occurs at thereceiver limiter, where it is treated as noise and rejected.– When both signals are nearly equal in strength, or are fadingindependently, the receiver may switch from one to the other.• Bandwith: BAM=2 x fm BFM >= 2 x fm use Carson’s Rule• The Receiver IF amplifier is change to a LimitingAmplifier for FM– FM rejects amplitude noise such as lightening and man madenoise• The FM demodulator may be a PLL, Ratio Detector,Foster Sealy Discriminator, or slope detector.

Trang 1

Lecture 28-29 FM- Frequency Modulation

PM - Phase Modulation

EE445-10

FM and PM

3

FM and PM

for FM:

Figure 5–8 Angle modulator circuits RFC = radio-frequency choke.

Trang 2

FM and PM

Figure 5–9 FM with a sinusoidal baseband modulating signal.

7

FM and PM

FM and PM differences

⇒

)

instantaneous frequency deviation from the carrier is proportional to m(t)

∫− ∞

D

PM:

FM:

⇒

=

)

volt

Hz K

D

volt

radians K

D

f f

p p

⇒

=

⇒

=

Modulation Constants

Trang 3

FM from PM

PM from FM

FM from PM

PM from FM

11

FM and PM Signals Maximum phase deviation in PM:

Maximum frequency deviation in FM:

Example Let

For PM For FM

Define the modulation indices:

Trang 4

Example Define the modulation indices:

Sine Wave Example

Then

15

Spectrum Characteristics of FM

• FM/PM is exponential modulation Let

( ( 2 sin( 2 )))

Re

)) 2 sin(

2 cos(

) (

t t

j c

m c

c

m c

e A

t f t

f A

t u

π β π

+

=

+

=

) 2 sin(

t β πm

φ =

we may therefore use the Fourier series

Spectrum Characteristics of FM

• FM/PM is exponential modulation

( ( 2 sin( 2 )))

Re

)) 2 sin(

2 cos(

) (

t t

j c

m c

c

m c

e A

t f t

f A

t c

π β π

+

=

+

=

Trang 5

Spectrum Characteristics with

Sinusoidal Modulation

19

Figure 5–11 Magnitude spectra for FM or PM with sinusoidal modulation for

various modulation indexes.

stop 3/29

Trang 6

Lecture 29 FM- Frequency Modulation

PM - Phase Modulation

(continued)

EE445-10

Narrowband FM

•Same Bandwidth as AM

•Using Eulers identity, and φ(t)<<1:

Notice the sidebands are “sin”, not “cos” as in AM

23

Narrowband FM as a Phaser

AM

NBFM

Wideband FM from Narrowband

FM (s(t))n

s(t)

ωc

βFM

s(t)

•The output bandwidth increases according to Carson’s Rule

Trang 7

Effective Bandwidth- Carson’s Rule

for Sine Wave Modulation

Where β is the modulation index

27

Trang 8

When m(t) is a sum of sine waves

31

Sideband Power Signal Amplitude: Ac := 1V

Modulating frequency: fm := 1KHz Carrier peak deveation: Δf := 2.4KHz Modulation index: β Δf

fm :=

β = 2.4

Reference equation: x t ( )

∞

−

∞

n

Ac Jn n β ⋅ ( , ) ⋅ cos ⎡⎣(ωc+ n ⋅ ωm)⋅ t ⎤⎦

∑

=

Power in the signal: Pc Ac

2

2 1 ⋅ Ω := Pc = 0.5 W

Carsons rule bandwidth: BW := 2 ⋅ ( β + 1 ) ⋅ fm

BW 6.8 × 1031 s

= Order of significant sidbands predicted by Carsons rule: n := round ( β + 1 )

n = 3

Power as a function of number of sidebands: Psum k ( )

k

− k n

Ac Jn n β ⋅ ( , )

2 1 ⋅ Ω

∑

= :=

Percent of power predicted by Carsons rule: Psum n ( )

Pc ⋅100=99.118

Sideband Power

0 50

100

PERCENT OF TOTAL POWER

Psum k ( )

Pc ⋅100

k

Trang 9

Sideband Power

k := 0 10

Pk:=( )J 2

1 n

j

Pj

∑

=

⋅

J

0

1

2

3

4

5

6

7

8

9

10

-3

2.508·10

0.52

0.431

0.198

0.064

0.016

-3

3.367·10

-4

5.927·10

-5

9.076·10

-5

1.23·10

-6

1.496·10

0 0 1 2 3 4 5 6 7 8 9 10

-6

6.288·10 0.271 0.186 0.039

-3

4.135·10

-4

2.638·10

-5

1.134·10

-7

3.513·10

-9

8.237·10

-10

1.513·10

-12

2.238·10

=

Sideband Power

n := 1

Vj := Jn j( ), β

Uj :=( )Vj 2

1 n

j Uj

∑

=

⋅

V

0.998

0.05

×

⎛⎜

⎜

⎜⎝

⎞⎟

⎟

⎟⎠

0.995

× 4.335 × 10 −10

× 0

⎛⎜

⎜

⎜⎝

⎞⎟

⎟

⎟⎠

=

35

Sideband Power

Wj := Jn j( ), β

Xj :=( )Wj 2

1 n

j Xj

∑

=

⋅

W

0.912 0.287 0.044 4.4 × 10 −3

×

⎛

⎜

⎝

⎞

⎟

⎠

0.832 0.082

× 1.936 × 10 −5 1.099 × 10 −7

×

⎛⎜

⎜

⎜⎝

⎞⎟

⎟

⎟⎠

=

Modulation_index 7.9 = Bandwidth 18 =

f := fc − ( n + 1 ) Fm ⋅ ,(fc n Fm− ⋅ ) ⎡⎣fc+ ( n + 1 ) Fm ⋅ ⎤⎦

B f ( ) := δ f f ⎡⎣ ,c+ ( n + 0 ) Fm ⋅ ⎤⎦ + δ f f(,c− n Fm ⋅ )

Si f ( ) Ac J0 M ( ( ) ) ⋅ δ f f( ),c

1 n k

Jn k M ( , ) ⋅ δ f f ⎡⎣ ,(c+ k Fm ⋅ )⎤⎦ + ( − 1 )k⋅ Jn k M ( , ) ⋅ δ f f ⎡⎣ ,(c− k Fm ⋅ )⎤⎦

∑

= +

⎡

⎢

⎤

⎥

⋅ :=

Modulation_index M :=

n = 9 Bandwidth 2 n := ⋅ Fm ⋅

2 * n is the number of significant sidebands per Carsons rule

n := round M ( + 1 )

M x 10 :=

Modulating frequency- single sinewave

Fm := 100

fc := 0 10 ⋅ 4

Ac := 1

79

FM/PM modulation index : set to π/2 for peak phase dev of π/2

set to Δf/fm for frequency modulation spectru

is the same for sinewave modulation.

filename: fmsidebands.mcd avo 09/21/04 last edit date:2/27/07

Trang 10

M=.4, Sideband Level =M/2 for Narrowband FM

0.2 0 0.2 0.4 0.6 0.8

Spectrum

Single Sided Spectrum

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J0

1st Sidebands J1

2nd Sidebands J2

Bessel Functions

Modulation_index

M=.9, Sideband Level =M/2 for Narrowband FM

0.5 0 0.5 1

Spectrum

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J0

1st Sidebands J1

2nd Sidebands J2

Bessel Functions

Modulation_index

39

M=2.4, Carrier Null

0.6 0.4 0.2 0 0.2 0.4 0.6

Spectrum

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

Carrier J0 1st Sidebands J1 2nd Sidebands J2

Bessel Functions Modulation_index

M=3.8, first sideband null

0.6 0.4 0.2 0 0.2 0.4 0.6

Spectrum

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

Carrier J0 1st Sidebands J1 2nd Sidebands J2 Bessel Functions Modulation_index

Trang 11

M=5.1, second sideband null

8 6 4 2 0 2 4 6 8 0.4

0.2 0 0.2 0.4

Spectrum

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Carrier J0

1st Sidebands J1

2nd Sidebands J2

Bessel Functions

Modulation_index

Power vs BW, M=0.1

second term includes power in +Jn and power in -Jn, i.e the upper and lower sideband pairs

P M n ( , ) J0 M( )

2 2

1 n k

Jn k M ( , )2

∑

=

⎛

⎜

⎞

⎟

:=

1 1.2 1.4 1.6 1.8

99.99965

99.9997

99.99975

99.9998

99.99985

99.9999

99.99995

% power vs bandwidth

Number of Sideband pairs

P M k ( , )

Ac 2

2

100

⋅

n

k

M = 0.1

Fm = 1 Hz Bandwidth 2 = Hz

P M n ( , )

Ac2 2 100

⋅ = 100

43

Power vs BW, M=0.9

98 98.5 99 99.5

100 % power vs bandwidth

P M k ( , )

Ac2 2 100

⋅

n

k

M = 0.9

P M n ( , )

Ac2 2

100

⋅ = 99.958

Power vs BW, M=2.4

50 60 70 80 90

100 % power vs bandwidth

P M k ( , )

Ac2 2 100

⋅

n

k

M = 2.4

P M n ( , )

Ac2 2

100

⋅ = 99.945

Trang 12

Figure 5–16 Angle-modulated system with preemphasis and deemphasis.

47

AM vs FM

• FM capture effect: A phenomenon, associated with FM reception, in which only the stronger of two signals at or near the same frequency will be demodulated

– The complete suppression of the weaker signal occurs at the receiver limiter, where it is treated as noise and rejected – When both signals are nearly equal in strength, or are fading independently, the receiver may switch from one to the other

• The Receiver IF amplifier is change to a Limiting Amplifier for FM

– FM rejects amplitude noise such as lightening and man made noise

• The FM demodulator may be a PLL, Ratio Detector, Foster Sealy Discriminator, or slope detector

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